We use the quasi-adiabatic approximation since the time scales of the superradiant instability and the GW emission are much longer than the dynamical time scale of the BH [1]}, [2]}. Without accretion, the evolution equations are,
\(& \dot{M} = - \sum _{nlm} 2 M_{s}^{(nlm)} \omega _{I}^{(nlm)}, \\& \dot{J} = - \sum _{nlm} 2 m M^{(nlm)}_{s} \omega _{I}^{(nlm)} /\omega _R^{(nlm)}, \\& \dot{M}_s^{(nlm)} = 2 M_{s}^{(nlm)} \omega _{I}^{(nlm)} - \dot{E}_\mathrm {GW}^{(nlm)}, \\& \dot{J}_{s}^{(nlm)} = 2 m M^{(nlm)}_{s} \omega _{I}^{(nlm)} /\omega _R^{(nlm)} - m \dot{E}_\mathrm {GW}^{(nlm)}/\omega _R^{(nlm)},\) 
